Precalculus Right Triangle and Law of Sines/Cosines Study Guide

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Q1. Find the exact value of each of the six trigonometric functions of the angle P.

Background

Topic: Right Triangle Trigonometry

This question tests your understanding of how to find the sine, cosine, tangent, cosecant, secant, and cotangent of a given angle in a right triangle.

Key Terms and Formulas:

Sine: $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$
Cosine: $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$
Tangent: $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$
Cosecant: $\csc \theta = \frac{1}{\sin \theta}$
Secant: $\sec \theta = \frac{1}{\cos \theta}$
Cotangent: $\cot \theta = \frac{1}{\tan \theta}$

Step-by-Step Guidance

Identify the side lengths of the triangle associated with angle $P$ (opposite, adjacent, hypotenuse).
Write the ratios for $\sin P$, $\cos P$, and $\tan P$ using the side lengths.
Express $\csc P$, $\sec P$, and $\cot P$ as the reciprocals of the primary functions.
Simplify each ratio to its exact value (in simplest radical form if necessary).

Try solving on your own before revealing the answer!

Q2. Find $\cos \alpha$ when $a = 13$ and $c = 6$ in a right triangle.

Background

Topic: Right Triangle Trigonometry

This question asks you to find the cosine of an angle in a right triangle given two side lengths.

Key Terms and Formulas:

Cosine: $\cos \alpha = \frac{\text{adjacent}}{\text{hypotenuse}}$
Pythagorean Theorem: $a^2 + b^2 = c^2$ (if needed to find the missing side)

Step-by-Step Guidance

Identify which sides correspond to $a$ (adjacent or opposite) and $c$ (hypotenuse) relative to angle $\alpha$.
If necessary, use the Pythagorean Theorem to find the missing side.
Write the cosine ratio for $\alpha$ using the appropriate side lengths.
Simplify the ratio to its exact value.

Try solving on your own before revealing the answer!

Q3. Given $a = 4$, $\alpha = 35^\circ$, find $b$, $c$, and $\beta$ in the right triangle.

Background

Topic: Solving Right Triangles

This question requires you to use trigonometric ratios and triangle properties to find missing sides and angles.

Key Terms and Formulas:

Sine: $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$
Cosine: $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$
Tangent: $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$
Triangle angle sum: $\alpha + \beta = 90^\circ$ (for right triangles)

Step-by-Step Guidance

Use the given $a$ (usually the side opposite $\alpha$) and $\alpha$ to set up trigonometric ratios for $b$ and $c$.
Use $\sin \alpha$ or $\cos \alpha$ to solve for $c$ (hypotenuse).
Use $\cos \alpha$ or $\tan \alpha$ to solve for $b$ (adjacent side).
Find $\beta$ by subtracting $\alpha$ from $90^\circ$.

Try solving on your own before revealing the answer!

Q4. Given $a = 4$, $b = 2$, find $c$, $\alpha$, and $\beta$ in the right triangle.

Background

Topic: Solving Right Triangles

This question asks you to find the missing side and angles of a right triangle given two sides.

Key Terms and Formulas:

Pythagorean Theorem: $a^2 + b^2 = c^2$
Inverse trigonometric functions: $\alpha = \arcsin(\frac{a}{c})$, $\beta = \arccos(\frac{b}{c})$, etc.

Step-by-Step Guidance

Use the Pythagorean Theorem to solve for $c$ (hypotenuse).
Use $\sin \alpha = \frac{a}{c}$ or $\tan \alpha = \frac{a}{b}$ to find $\alpha$.
Use $\sin \beta = \frac{b}{c}$ or $\tan \beta = \frac{b}{a}$ to find $\beta$.
Check that $\alpha + \beta = 90^\circ$.

Try solving on your own before revealing the answer!

Q5. Find the exact value of $\sin 20^\circ \cos 70^\circ$ (do not use a calculator).

Background

Topic: Trigonometric Identities

This question tests your ability to use product-to-sum identities to simplify trigonometric expressions.

Key Terms and Formulas:

Product-to-sum identity: $\sin A \cos B = \frac{1}{2}[\sin(A+B) + \sin(A-B)]$

Step-by-Step Guidance

Identify $A = 20^\circ$ and $B = 70^\circ$.
Apply the product-to-sum identity to rewrite $\sin 20^\circ \cos 70^\circ$.
Simplify the resulting angles inside the sine functions.
Express the answer in exact form (no decimals).

Try solving on your own before revealing the answer!

Q6. Simplify $\tan 35^\circ - 2 \cot 55^\circ$ (do not use a calculator).

Background

Topic: Trigonometric Identities and Simplification

This question asks you to use cofunction identities and basic trigonometric relationships to simplify the expression.

Key Terms and Formulas:

Cofunction identity: $\cot(90^\circ - x) = \tan x$
$\cot x = \frac{1}{\tan x}$

Step-by-Step Guidance

Rewrite $\cot 55^\circ$ in terms of $\tan$ using the cofunction identity.
Substitute this into the expression and combine like terms.
Simplify the expression as much as possible.

Try solving on your own before revealing the answer!

Q7. Simplify $\tan 75^\circ - 2 \cot 15^\circ$ (do not use a calculator).

Background

Topic: Trigonometric Identities and Simplification

This question tests your ability to use angle sum identities and cofunction identities to simplify trigonometric expressions.

Key Terms and Formulas:

Angle sum identity: $\tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$
Cofunction identity: $\cot(90^\circ - x) = \tan x$

Step-by-Step Guidance

Express $\tan 75^\circ$ using the angle sum identity ($75^\circ = 45^\circ + 30^\circ$).
Rewrite $\cot 15^\circ$ in terms of $\tan$ using the cofunction identity.
Substitute and simplify the expression step by step.

Try solving on your own before revealing the answer!

Q8. A 22-foot extension ladder leaning against a building makes a $70^\circ$ angle with the ground. How far up the building does the ladder touch?

Background

Topic: Right Triangle Applications

This question involves applying right triangle trigonometry to a real-world scenario.

Key Terms and Formulas:

Sine: $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$

Step-by-Step Guidance

Draw a right triangle representing the ladder, wall, and ground.
Identify the side opposite the angle ($70^\circ$) as the height up the building, and the hypotenuse as the ladder (22 ft).
Set up the equation using $\sin 70^\circ = \frac{\text{height}}{22}$.
Rearrange to solve for the height.

Try solving on your own before revealing the answer!

Q9. A 12 meter flagpole casts a 9 meter shadow. Find the angle of elevation to the sun from the ground.

Background

Topic: Right Triangle Applications

This question asks you to use trigonometric ratios to find an angle given two sides of a right triangle.

Key Terms and Formulas:

Tangent: $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$
Inverse tangent: $\theta = \arctan\left(\frac{\text{opposite}}{\text{adjacent}}\right)$

Step-by-Step Guidance

Identify the height of the flagpole as the side opposite the angle, and the shadow as the adjacent side.
Set up the equation $\tan \theta = \frac{12}{9}$.
Take the inverse tangent of both sides to solve for $\theta$.

Try solving on your own before revealing the answer!

Q10. A kite is caught at the top of a tree. The string is 100 feet long and makes a $75^\circ$ angle with the ground. How tall is the tree?

Background

Topic: Right Triangle Applications

This question involves using trigonometric ratios to find the height of an object given the hypotenuse and an angle.

Key Terms and Formulas:

Sine: $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$

Step-by-Step Guidance

Draw a right triangle with the string as the hypotenuse and the height of the tree as the side opposite the $75^\circ$ angle.
Set up the equation $\sin 75^\circ = \frac{\text{height}}{100}$.
Rearrange to solve for the height.

Try solving on your own before revealing the answer!

Q11. Solve the triangle: $\gamma = 85^\circ$, $\alpha = 50^\circ$, $b = 3$.

Background

Topic: Law of Sines (SAA/ASA/SSA Triangles)

This question asks you to solve a triangle using the Law of Sines, given two angles and a side.

Key Terms and Formulas:

Law of Sines: $\frac{a}{\sin \alpha} = \frac{b}{\sin \beta} = \frac{c}{\sin \gamma}$
Triangle angle sum: $\alpha + \beta + \gamma = 180^\circ$

Step-by-Step Guidance

Find the missing angle $\beta$ using the triangle angle sum.
Set up the Law of Sines to solve for the missing sides $a$ and $c$.
Plug in the known values and solve for one missing side at a time.

Try solving on your own before revealing the answer!

Q12. Solve the triangle: $\beta = 40^\circ$, $b = 2$, $c = 3$.

Background

Topic: Law of Sines (SSA Triangle)

This question requires you to use the Law of Sines to solve for the missing sides and angles of a triangle given two sides and a non-included angle.

Key Terms and Formulas:

Law of Sines: $\frac{a}{\sin \alpha} = \frac{b}{\sin \beta} = \frac{c}{\sin \gamma}$

Step-by-Step Guidance

Set up the Law of Sines using the given values to solve for one of the unknown angles.
Use the triangle angle sum to find the remaining angle.
Use the Law of Sines again to solve for the missing side.

Try solving on your own before revealing the answer!

Q13. Solve the triangle: $\alpha = 70^\circ$, $a = 3$, $b = 7$.

Background

Topic: Law of Sines (SSA Triangle)

This question asks you to use the Law of Sines to solve for the missing sides and angles of a triangle given two sides and an angle.

Key Terms and Formulas:

Law of Sines: $\frac{a}{\sin \alpha} = \frac{b}{\sin \beta} = \frac{c}{\sin \gamma}$

Step-by-Step Guidance

Set up the Law of Sines to solve for $\beta$.
Use the triangle angle sum to find $\gamma$.
Use the Law of Sines to solve for $c$.

Try solving on your own before revealing the answer!

Q14. Find the missing part of the triangle: $a = 26$, $\alpha = 4^\circ$, $\beta = 5^\circ$.

Background

Topic: Law of Sines (SAA Triangle)

This question asks you to use the Law of Sines to find the missing side or angle in a triangle given two angles and a side.

Key Terms and Formulas:

Law of Sines: $\frac{a}{\sin \alpha} = \frac{b}{\sin \beta} = \frac{c}{\sin \gamma}$
Triangle angle sum: $\alpha + \beta + \gamma = 180^\circ$

Step-by-Step Guidance

Find $\gamma$ using the triangle angle sum.
Set up the Law of Sines to solve for the missing side(s).
Plug in the known values and solve for the unknown(s).

Try solving on your own before revealing the answer!

Q15. Find the missing part of the triangle: $a = 3.44$, $\beta = 2^\circ$, $c = 2.3$.

Background

Topic: Law of Sines (SSA Triangle)

This question asks you to use the Law of Sines to find the missing sides or angles in a triangle given two sides and an angle.

Key Terms and Formulas:

Law of Sines: $\frac{a}{\sin \alpha} = \frac{b}{\sin \beta} = \frac{c}{\sin \gamma}$

Step-by-Step Guidance

Set up the Law of Sines to solve for one of the unknown angles.
Use the triangle angle sum to find the remaining angle.
Use the Law of Sines to solve for the missing side.

Try solving on your own before revealing the answer!

Q16. John wants to measure the height of a tree. He walks 100 feet from the base and looks up at a $33^\circ$ angle. The tree grows at an $83^\circ$ angle with respect to the ground. How tall is the tree?

Background

Topic: Law of Sines (Oblique Triangle Application)

This question involves using the Law of Sines to solve for the height of a tree that is not perpendicular to the ground.

Key Terms and Formulas:

Law of Sines: $\frac{a}{\sin \alpha} = \frac{b}{\sin \beta} = \frac{c}{\sin \gamma}$
Triangle angle sum: $\alpha + \beta + \gamma = 180^\circ$

Step-by-Step Guidance

Draw a diagram and label all known sides and angles.
Use the triangle angle sum to find the third angle.
Set up the Law of Sines to solve for the side representing the tree's height.

Try solving on your own before revealing the answer!

Q17. Solve the triangle: $\beta = 20^\circ$, $a = 2$, $c = 5$.

Background

Topic: Law of Cosines (SAS/SSS Triangles)

This question asks you to use the Law of Cosines to solve for the missing sides and angles of a triangle given two sides and an angle.

Key Terms and Formulas:

Law of Cosines: $c^2 = a^2 + b^2 - 2ab \cos \gamma$
Law of Sines: $\frac{a}{\sin \alpha} = \frac{b}{\sin \beta} = \frac{c}{\sin \gamma}$

Step-by-Step Guidance

Use the Law of Cosines to solve for the missing side $b$.
Use the Law of Sines to solve for the remaining angles.
Check that the sum of the angles is $180^\circ$.

Try solving on your own before revealing the answer!

Q18. Solve the triangle: $a = 4$, $b = 3$, $c = 6$.

Background

Topic: Law of Cosines (SSS Triangle)

This question asks you to use the Law of Cosines to find all angles of a triangle given all three sides.

Key Terms and Formulas:

Law of Cosines: $c^2 = a^2 + b^2 - 2ab \cos \gamma$

Step-by-Step Guidance

Use the Law of Cosines to solve for one angle (e.g., $\gamma$).
Repeat for the other two angles using the Law of Cosines or Law of Sines.
Check that the sum of the angles is $180^\circ$.

Try solving on your own before revealing the answer!

Q19. Solve the triangle: $a = 7.20$, $b = 8.89$, $\gamma = 113.2^\circ$.

Background

Topic: Law of Cosines (SAS Triangle)

This question asks you to use the Law of Cosines to solve for the missing side and angles of a triangle given two sides and the included angle.

Key Terms and Formulas:

Law of Cosines: $c^2 = a^2 + b^2 - 2ab \cos \gamma$
Law of Sines: $\frac{a}{\sin \alpha} = \frac{b}{\sin \beta} = \frac{c}{\sin \gamma}$

Step-by-Step Guidance

Use the Law of Cosines to solve for the missing side $c$.
Use the Law of Sines to solve for one of the remaining angles.
Find the last angle using the triangle angle sum.

Try solving on your own before revealing the answer!

Q20. Find the missing part of the triangle: $a = 15$, $b = 17$, $c = 13$.

Background

Topic: Law of Cosines (SSS Triangle)

This question asks you to use the Law of Cosines to find the angles of a triangle given all three sides.

Key Terms and Formulas:

Law of Cosines: $c^2 = a^2 + b^2 - 2ab \cos \gamma$

Step-by-Step Guidance

Use the Law of Cosines to solve for one angle (e.g., $\gamma$).
Repeat for the other two angles using the Law of Cosines or Law of Sines.
Check that the sum of the angles is $180^\circ$.

Try solving on your own before revealing the answer!

Q21. The distance from home plate to dead center field in US Cellular Field is 400 feet. A baseball diamond is a square with a distance from home plate to first base of 90 feet. How far is it from first base to dead center field?

Background

Topic: Law of Cosines (Application Problem)

This question asks you to model a real-world scenario as a triangle and use the Law of Cosines to find the unknown distance.

Key Terms and Formulas:

Law of Cosines: $c^2 = a^2 + b^2 - 2ab \cos \gamma$
Understanding the geometry of a baseball diamond (square, right angles)

Step-by-Step Guidance

Draw a diagram of the baseball diamond and label all known distances.
Identify the triangle formed by home plate, first base, and dead center field.
Determine the included angle at first base (likely $90^\circ$).
Set up the Law of Cosines to solve for the unknown distance from first base to dead center field.